Q. 94.3( 362 Votes )

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that DP = BQ (see Fig. 8.20). Show that:



(i) Δ APD Δ CQB

(ii) AP = CQ

(iii) Δ AQB Δ CPD

(iv) AQ = CP

(v) APCQ is a parallelogram

Answer :


(i) In ΔAPD and ΔCQB,

ADP = CBQ (Alternate interior angles for BC || AD)

AD = CB (Opposite sides of parallelogram ABCD)

DP = BQ (Given)

Two sides and included angle (SAS) Definition: Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAPD ΔCQB (Using SAS congruence rule)


(ii) As we had observed that,

ΔAPD ΔCQB

AP = CQ (CPCT)


(iii) In ΔAQB and ΔCPD,

ABQ = CDP (Alternate interior angles for AB || CD)

AB = CD (Opposite sides of parallelogram ABCD)

BQ = DP (Given)

Two sides and included angle (SAS) Definition: Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles.

ΔAQB ΔCPD (Using SAS congruence rule)


(iv) As we had observed that,


ΔAQB ΔCPD,


AQ = CP (CPCT)


(v) From the result obtained in (ii) and (iv),

AQ = CP and AP = CQ

Since,


Opposite sides in quadrilateral APCQ are equal to each other,


APCQ is a parallelogram

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Critical Thinking Problems on QuadrilateralsCritical Thinking Problems on QuadrilateralsCritical Thinking Problems on Quadrilaterals44 mins
Quiz | Properties of ParallelogramQuiz | Properties of ParallelogramQuiz | Properties of Parallelogram31 mins
Extras on QuadrilateralsExtras on QuadrilateralsExtras on Quadrilaterals40 mins
Smart Revision | QuadrilateralsSmart Revision | QuadrilateralsSmart Revision | Quadrilaterals43 mins
Quiz | Basics of QuadrilateralsQuiz | Basics of QuadrilateralsQuiz | Basics of Quadrilaterals36 mins
RD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma |  Extra Qs. of Cyclic QuadrilateralsRD Sharma | Extra Qs. of Cyclic Quadrilaterals31 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses